The SRR can be modeled as an RLC circuit with external excitation and the voltage equation can be expressed by \(-L \frac{\, dI}{\, dt}-R_S-V_D = \varepsilon (t)\) , where \(I\)  is the current in the resonator, \(L\)  is the inductance of the resonator determined by the ring geometry, \(R_S\)  is the resistance, and \(\varepsilon \)  is the driven term provided by the loop antenna in the experiment. For small excitations, \(I_D\)  can be neglected so the current can be estimated by \(I \approx \, dQ_D/\, dt\) . The equation of motion is now \(\frac{\, d^2q}{\, dt^2}+ \gamma \frac{\, dq}{\, dt} + \omega _0^2 V_D = -\omega _0^2 \varepsilon (t)\) , where \(\omega _0^2 = 1/(LC_0)\)  and \(\gamma = \omega _0^2 R_S C_0\) . Expand the restoring term \(V_D\)  by the Taylor series for small oscillations (the oscillation amplitude satisfies \((1-M)|q|<V_p\) ) and omit the higher order terms, \(V_D(q) \approx q - \frac{M}{2V_p}q^2 + \frac{M(2M-1)}{6V_p^2}q^3\) . Assume harmonic excitation so that \(\varepsilon (t) = f \cos (\omega t)\) , where \(f\)  is the excitation amplitude and \(\omega \)  is the excitation frequency, the equation of motion is further estimated by \(\frac{\, d^2q}{\, dt^2}+ \gamma \frac{\, dq}{\, dt} + \omega _0^2 q + \alpha q^2 +\beta q^3 = -\omega _0^2 f \cos (\omega t)\) , where \(\alpha = - \frac{\omega _0^2 M}{2 V_p}\) , \(\beta = \frac{\omega _0^2 M (2M-1)}{6V_p^2}\) . This is now a nonlinear driven oscillator problem [1]}.
